THE RESTRICTION PROBLEM AND THE TOMAS-STEIN
THEOREM
DENNIS KRIVENTSOV
Abstract. E. M. Stein’s restriction problem for Fourier transforms is a deep
and only partially solved conjecture in harmonic analysis. Below we state
the problem and the Tomas-Stein theorem, which solves a particularly useful
case of the conjecture. We then introduce the Fourier transform of complexvalued measures and the stationary phase method as tools used in the proof
of the Tomas-Stein theorem. We give a proof of the theorem, and then turn to
applications in deriving the Strichartz estimate for the Schroedinger equation.
Contents
1. Introduction
2. Fourier transforms of measures
3. Stationary phase approximations
4. Proof of the Tomas-Stein theorem
5. Strichartz Estimates
Acknowledgments
References
1
2
4
6
9
10
11
1. Introduction
We assume the reader is familiar with basic facts about the Lp spaces, L1 and
L Fourier transforms, and convolutions. These can be found in most analysis
textbooks, such as [2] or [1]. Given a function f : Rn → C we will use the following
notation and normalization to denote the Fourier transform (when defined):
2
(1.1)
fˆ(ξ) =
Z
e−2πix·ξ f (x)dx
Rn
and the inverse Fourier transform:
(1.2)
fˇ(x) =
Z
e2πiξ·x f (ξ)dξ.
Rn
The restriction problem asks when an inequality of the form
(1.3)
kfˆ|S n−1 kLq (S n−1 ) ≤ Ckf kLp
holds, where S n−1 is the unit sphere and the constant depends only on p, q, and n.
When q = 2, the best possible result is given by the following theorem:
Date: August 3, 2009.
1
2
DENNIS KRIVENTSOV
0
Theorem 1.4 (Tomas-Stein). Let f ∈ Lp with p0 ≤
Ckf kLp0 , where C depends only on n and p0 .
2n+2
n+3 .
Then kfˆ|S n−1 kL2 (S n−1 ) ≤
We will give a proof for the region p0 < 2n+2
n+3 . The argument for the endpoint is
quite different, using a complex interpolation method; it can be found in [3]. The
next section introduces the Fourier transform of a measure and the convolution of
a measure with a function. Section 3 derives estimates for the decay of the Fourier
transform of the surface measure of the sphere. Section 4 fills in the remaining
details of the proof. Section 5 gives a useful application to differential equations.
Sections 2-4 follow the exposition in [4] while 5 follows [3].
2. Fourier transforms of measures
We will want to rephrase the statement of the Tomas-Stein theorem in terms
of the surface measure of a sphere. To do this, we will first prove some general
statements about complex-valued measures.
Definitions 2.1. Let µ be a complex-valued measure on Rn with finite total variation. Then µ̂ : Rn → C, the Fourier transform of the measure, is given by
Z
µ̂(ξ) =
e−2πix·ξ dµ(x).
Rn
n
Let ϕ : R → C be a Schwartz function (i.e. smooth, rapid decay, and rapid
decay for all derivatives). Then the convolution of ϕ and µ is
Z
ϕ ∗ µ(x) =
ϕ(x − y)dµ(y).
Rn
It is clear that both the Fourier transform of µ and its convolution with ϕ are
bounded, since
Z
Z
|µ̂| = |
e−2πix·ξ dµ| ≤
|e−2πix·ξ |d|µ| = |µ|(Rn ) < ∞
Rn
Rn
by assumption. For the convolution,
Z
Z
|ϕ ∗ µ| = |
ϕ(x − y)dµ(y)| ≤
|ϕ(x − y)|d|µ|(y) ≤ (sup |ϕ|) |µ|(Rn ) < ∞,
Rn
Rn
using that Schwartz functions are bounded.
The following lemma states that the ordinary interaction between convolutions
and the Fourier transform carry over to measures.
Lemma 2.2. Let µ and ϕ be as above, and ν is another finite measure. Then we
have:
c = ϕ̂ ∗ µ
(1) ϕµ̌
(2) ϕµ
Rc = ϕ̂ ∗Rµ̂
(3) µ̂dν = ν̂dµ.
Proof. These can be derived from Fubini’s theorem. We start with the duality
relation (3):
Z
ZZ
ZZ
Z
µ̂dν =
e−2πix·ξ dµ(x)dν(ξ) =
e−2πix·ξ dν(ξ)dµ(x) = ν̂dµ,
with Fubini’s theorem clearly justified since both measures are finite.
THE RESTRICTION PROBLEM AND THE TOMAS-STEIN THEOREM
3
To see (1), fix ξ ∈ Rn . Consider e−2πix·(ξ−y) ϕ(x) as a function from Rn × Rn
under the product measure dx × dµ(y). It is integrable by Tonelli’s theorem:
Z
ZZ
|e−2πix·(ξ−y) ϕ(x)|dx×d|µ|(y) =
|ϕ(x)|dxd|µ|(y) ≤ kϕkL1 ·|µ|(Rn ) < ∞.
Rn ×Rn
Rn ×Rn
We can therefore apply Fubini’s theorem to the following:
Z
Z
Z
−2πix·ξ
−2πix·ξ
2πiy·x
c
e
ϕµ̌(ξ) = e
ϕ(x)µ̌(x)dx = e
ϕ(x)
dµ(y) dx
ZZ
Z
=
e−2πix·(ξ−y) ϕ(x)dxdµ(y) = ϕ̂(ξ − y)dµ(y) = φ̂ ∗ µ(ξ).
(2) also falls out from Fubini’s theorem, this time using the integrability of ϕ̂ to
justify switching the order of integration:
Z
Z Z
e−2πiξ·(x−y) dµ(ξ) ϕ̂(y)dy
ϕ̂ ∗ µ̂(x) = µ̂(x − y)ϕ̂(y)dy =
Z Z
Z
=
e2πiξ·y ϕ̂(y)dy e−2πiξ·x dµ(ξ) = ϕ(ξ)e−2πiξ·x dµ(ξ) = ϕµ.
c
We have already proven most of the next lemma:
Lemma 2.3. Let µ be a finite measure and f and g Schwartz functions. Then
Z
Z
¯ = (µ̂ ∗ ḡ) · f dx.
fˆĝdµ
Proof. The only new fact needed is that ĝ¯ = ḡˇ. This follows from the fact that the
integral of the conjugate is the conjugate of the integral:
Z
Z
Z
ĝ¯ = e−2πix·ξ g(x)dx = e−2πix·ξ g(x)dx = e2πix·ξ g(x)dx = ḡˇ.
Now we can simply compute, using first duality (Lemma 2.2(3)) and then Lemma
2.2(2):
Z
Z
Z
Z
Z
c̄ = f · (ĝˆ¯ ∗ µ̂)dx = f · (ḡˆˇ ∗ µ̂)dx = f · (ḡ ∗ µ̂)dx.
¯ = f · ĝµdx
fˆĝdµ
We are now in a position to make the major simplifying restatement of the
Tomas-Stein theorem. This is known as the T T ∗ approach for this reason: the
Tomas-Stein theorem as stated above asserts the boundedness of the operator T ∗
from Lp to L2 (S n−1 ) which is given by taking the Fourier transform and restricting
it to the sphere. We claim this is equivalent to its adjoint T being bounded and
the composition T T ∗ being bounded.
Theorem 2.4. Let p and p0 satisfy
1
1
+
= 1.
p p0
Let µ be a finite measure on Rn . Then the following are equivalent:
(1) kfc
µkLp ≤ Ckf kL2 (µ) for all f ∈ L2 (µ)
(2) kĝkL2 (µ) ≤ CkgkLp0 for all Schwartz functions g
4
DENNIS KRIVENTSOV
(3) kµ̂ ∗ hkLp ≤ C 2 khkLp0 for all Schwartz functions h.
Proof. The proof uses the following basic fact about dual Banach spaces (self-dual,
in this case):
Z
kĝkL2 (µ) =
sup
| ĝf dµ|.
f ∈L2 (µ),kf kL2 (µ) =1
Then if (1) holds, we can use the duality relation (Lemma 2.2(c)):
Z
Z
(duality)
sup
| ĝf dµ| =
sup
| fc
µ · gdx|
kf kL2 (µ) =1
kf kL2 (µ) =1
≤
(Holder)
sup
kf kL2 (µ) =1
≤
(1)
sup
kf kL2 (µ) =1
kfc
µkLp kgkLp0
Ckf kL2 (µ) kgkLp0
= CkgkLp0 ,
which is (2). Conversely, if we assume (2) by the same reasoning we have
Z
Z
c
f µ · gdx = ĝf dµ ≤ kĝkL2 (µ) kf kL2 (µ) ≤ CkgkLp0 kf kL2 (µ) .
0
Since Schwartz functions are dense in Lp , we then have
Z
c
kf µkLp =
sup
| fc
µ · gdx| ≤
sup
CkgkLp0 kf kL2 (µ) = Ckf kL2 (µ) ,
g∈S,kgkLp0 =1
g∈S,kgkLp0 =1
which is (1) (S is the Schwartz space).
Next we show (2) and (3) are equivalent. If (3) holds, then
Z
Z
¯
2
kĥkL2 (µ) = ĥĥdµ = (µ̂ ∗ h̄) · hdx ≤ kµ̂ ∗ h̄kLp khkLp0 ≤ C 2 khk2Lp0 ,
which is the square of (2). The second equality is Lemma 2.3 with both functions
set to h, the next step is Holder’s inequality, and the last inequality is (3) by
assumption. Now assume (2) and let h be a Schwartz function. Then, using the
same tools,
Z
(dual spaces)
kµ̂ ∗ h̄kLp =
sup
| (µ̂ ∗ h̄) · gdx|
g∈S,kgkLp0 =1
Z
(Lemma 2.3)
= sup |
¯
ĝ ĥdµ|
g
(Cauchy-Schwartz)
≤ sup kĥkL2 (µ) kĝkL2 (µ)
g
(2)
≤ C 2 sup khkLp0 kgkLp0 = C 2 khkLp0 ,
g
which is (3).
3. Stationary phase approximations
The previous section rephrased the restriction problem in terms of the Fourier
transform of the surface of the sphere. In order to exploit this, we will need to
THE RESTRICTION PROBLEM AND THE TOMAS-STEIN THEOREM
5
estimate the decay of this function. First, we consider a more general question. Let
φ : Rn → C be smooth and α : Rn → C be smooth and of compact support. Let
Z
I(λ) = e−πiλφ(x) α(x)dx.
We wish to estimate the decay of |I(λ)| as λ → ∞. The following computation
shows that if such an estimate is independent of α, it is diffeomorphism invariant.
More precisely, if φ1 and φ2 are smooth functions with φ1 = φ2 ◦ G for some
diffeomorphism G,
Z
Z
Z
−1
e−πiλφ2 (x) α(x)dx = e−πiλφ1 (G x) α(x)dx = e−πiλφ1 (y) α(Gy)|J(G)|dy,
where J takes the determinant of the Jacobian matrix.
The decay on I is related to the degeneracy of φ. The simplest case of this is
when φ is nonstationary. The following is a general fact about smooth functions;
the proof is omitted.
Lemma 3.1 (Straightening). Let f be a smooth complex valued function on a neighborhood of p with ∇f (p) 6= 0. Then there is a diffeomorphism G of neighborhoods
of 0 and p with G(p) = 0 and f ◦ G(x) = f (p) + xn .
This immediately gives the following:
Theorem 3.2 (Nonstationary phase). Let φ be smooth on a neighborhood of p with
∇f (p) 6= 0. Then for α smooth and supported on a sufficiently small neighborhood
of p, for all N there is a CN (that depends on α and φ) such that |I(λ)| ≤ CN λ−N .
Proof. Let G : U → V be the diffeomorphism given by applying Lemma 3.1. Assume α is supported inside V . Then by the computation above,
Z
λen
)
|I(λ)| = | e−πiλ(φ(p)+yn ) α(Gy)|J(G)|dy| = C|α ◦ \
G|J(G)||(
2
where in the last step the phase e−πiλφ(p) was pulled out of the integral. But
the function α(Gy)|J(G)| is C0∞ , so its Fourier transform is Schwartz and can be
bounded as required.
The decay grows weaker for a nondegenerate critical point, but the proof of this
is somewhat harder. The replacement for Lemma 3.1 is the Morse lemma. The
calculation is more sophisticated and requires the Fourier transform of tempered
distributions, but the general format (reducing to a normal form by diffeomorphism
invariance) is the same. The result is stated below; the proof can be found in [4]
Theorem 3.3 (Stationary phase). Let φ be smooth on a neighborhood of p with
∇f (p) 6= 0. Assume the Hessian matrix Hφ (p) is invertible. Then for α smooth
and supported on a sufficiently small neighborhood of p, there is a C (that depends
on α and φ) such that |I(λ)| ≤ Cλ−n/2 .
We wish to apply this to the surface measure of the sphere, which from now on
will be denoted by σ. The first observation is that since σ has rotational symmetry,
so does σ̂. Therefore it is sufficient to estimate the decay of σ̂(λen ). Next, we cover
S n−1 by coordinate chartsp
as follows: the first is the inverse of q1 : Dn−1 (0, 1/2) →
n−1
S
given by q1 (x) = (x, 1 − |x|2 ) paired with its image, which is a neighborhood
of the north pole. The second is the inverse of q2 : Dn−1 (0, 1/2) → S n−1 given by
6
DENNIS KRIVENTSOV
p
q2 (x) = (x, − 1 − |x|2 ) paired with its image, which is a neighborhood of the south
pole. Dn−1 (p, r) = D(p, r) here means {x ∈ Rn−1 | |x − p| < r}. The remaining
charts (with similar maps {qj }kj=1 ) avoid the north and south poles.
Let ϕj be a partition of unity subordinate to the covering by charts. Now we
have
Z
σ̂(λen ) = e−2πiλxn dσ(x)
=
k Z
X
e−2πiλxn ϕj (x)dσ(x)
j=1
√
Z
=
e
D(0,1/2)
+
k Z
X
j=3
−2πiλ
1−|y|2
ϕ1 (y,
p
p
1 − |y|2 )
1 − |y|2
√
Z
dy +
e
D(0,1/2)
2πiλ
1−|y|2
p
ϕ2 (y, − 1 − |y|2 )
p
dy
1 − |y|2
e−2πiλqj (y)·en ϕj ◦ qj (y)αj (y)dy
Uj
where Uj are the domains of the qj and the αj are smooth functions that depend
on qj . Consider the phase functions of the integrals in the sum. It is easy to see
that if ∇(en · λqj )(y) = 0, then the qj is stationary in the en direction; since qj
is a coordinate map, this happens only at two places on the sphere–the north and
south poles. But these are not in the support of ϕj for j > 2, so the phase there is
nonstationary everywhere. This means that Theorem 3.2 applies in a neighborhood
of every point, and so, by compactness, on the whole support of ϕj .
The other two integrals are each stationary at exactly one point (y = 0), and
there the Hessian can be computed to be −2I, which is invertible. Using Theorem
3.3 when y = 0 and Theorem 3.2 elsewhere and combining with the above, we get
|σ̂(λen )| ≤ Cλ−
n−1
2
.
Recalling from before that the Fourier transform of a finite measure is bounded
gives
Theorem 3.4. For all ξ ∈ Rn , we have
|σ̂(ξ)| ≤ C(1 + |ξ|)−
n−1
2
.
4. Proof of the Tomas-Stein theorem
The following is a well-known interpolation theorem we will use in the proof
below. Its proof can be found in [1].
Theorem 4.1 (Reisz-Thorin). Let T be a linear operator from Lp0 + Lp1 with
1 ≤ p0 ≤ p1 ≤ ∞. Assume
kT f kq0 ≤ A0 kf kp0
and
kT f kq1 ≤ A1 kf kp1
for some q0 , q1 ∈ [1, ∞]. Then if for some θ ∈ (0, 1),
1−θ
θ
1
=
+
p
p0
p1
and
1
1−θ
θ
=
+ ,
q
q0
q1
THE RESTRICTION PROBLEM AND THE TOMAS-STEIN THEOREM
7
then
kT f kq ≤ Aθ1 A1−θ
kf kp .
0
A consequence of this is Young’s inequality for convolutions:
Theorem 4.2 (Young). Let f ∈ Lp and g ∈ Lq with
such that 1r = p1 + 1q − 1,
kf ∗ gkr ≤ kf kp kgkq .
1
p
+
1
q
≥ 1. Then if we take r
Proof. Let T be the convolution operator T h = f ∗h. We will prove the inequalities
kT hkp ≤ kf kp khk1
(A)
and
kT hk∞ ≤ kf kp khkp0
(B)
0
0
where Lp and Lp are dual. Thinking of T as an operator from Lp + L1 to Lp + L∞ ,
then, we can apply Reisz-Thorin and set θ = 1 − pr . Then
1−θ
θ
p
1 p p
1
p1
1
1
1
+ 0 = + 0 1−
=
1− 0 + 0 =
+1− =
1
p
r
p
r
r
p
p
rp
p
q
and
1−θ
θ
1
+
= ,
p
∞
r
so
kf ∗ gkr = kT gkr ≤ kf k1−θ
kf kθp kgkq = kf kp kgkq
p
as desired.
To prove inequality (A), we use Reisz-Thorin again, this time fixing h ∈ L1 and
letting Rf = f ∗ h. Then if f is bounded, clearly we have
Z
|Rf | ≤ |f (x − y)||g(y)|dy ≤ kf k∞ kgk1 .
If f is in L1 , then
ZZ
ZZ
Z
kRf k1 ≤
|f (x − y)g(y)|dydx =
|f (x − y)|dx|g(y)|dy = kf k1 |g(y)|dy = kf k1 kgk1 .
We tus have R bounded L1 → L1 and L∞ → L∞ , so by Reisz-Thorin we have
kf ∗ hkp = kRf k ≤ kf kp khk1 as desired.
To finish the proof, we observe that inequality (B) is a trivial consequence of
Holder’s inequality.
Proof of Tomas-Stein. Recall that from Theorem 2.4, it suffices to show that for
2n+2
0
p
p0 < 2n+2
n+3 (or, equivalently, p > n−1 ) kσ̂ ∗ f kL ≤ Ckf kLp for all Schwartz
0
functions f . Since S is dense in Lp , this will imply Tomas-Stein as stated in
Theorem 1.4.
Next, let ψ be a smooth function Rn → [0, 1] that is 0 when |x| < 1/2 and 1
when |x| > 1. Then let φ(x) = ψ(2x) − ψ(x). Then when |x| > 1, φ(x) = 1 − 1 = 0,
and when |x| < 1/4, φ(x) = 0 − 0 = 0. Moreover, for |x| > 1,
X
φ(2−k x) = 1.
k≥0
8
DENNIS KRIVENTSOV
This means that we can write


∞
X
X
X
−k
−k


σ̂ = 1 −
φ(2 x) σ̂ +
(φ(2 x)σ̂) = K−∞ +
Kj ,
k≥0
j=0
k≥0
P
where we use K−∞ = 1 − j≥0 φ(2−j x) σ̂ and Kj = φ(2−j x)σ̂.
K−∞ has compact support by construction. This means that by Young’s inequality,
kK−∞ ∗ f kLp ≤ kK−∞ kLq kf kLp0
(4.3)
where 1q = 2 − p20 , which works because p0 > 2n+2
n+3 ≥ 2. As was noted earlier, σ̂ is
bounded, so kK−∞ kLq is finite and depends only on dimension.
n−1
In the previous section, we obtained the estimate |σ̂(ξ)| ≤ C(1 + |ξ|)− 2 , where
C depends only on n (Theorem 3.4). But because of the conditions on the support
n−1
of Kj , this gives kKj kL∞ ≤ C2−j 2 . Using a simple case of Young’s inequality,
kKj ∗ f kL∞ ≤ kKj kL∞ kf kL1 ≤ C2−j
(4.4)
n−1
2
kf kL1 .
At this point we need to make this observation:
σ(D(x, r)) ≤ Arn−1
(4.5)
where D(x, r) = {x ∈ Rn | |x| < r} and A depends only on n. This is clear since
S n−1 is an (n − 1) dimensional submanifold of Rn .
Next, because of the radial symmetry of the sphere σ, and so σ̂, is invariant
under reflection x → −x, and so σ̌(ξ) = σ̂(−ξ) = σ̂(ξ). Then we can apply Lemma
2.2(a) to Kj :
nj
\
K̂j = φ\
2−j σ̂ = φ2−j σ̌ = 2 φ̂2j ∗ σ,
where ga (x) = g(ax). Now, φ̂ is Schwartz, so for each N there is a constant CN
such that |φ̂(ξ)| ≤ CN (1 + |ξ|)−N . Letting N = n, we can perform the following
computation, decomposing the integral into a sum over annular regions:
Z
|K̂j (ξ)| = 2jn φ̂(2j (ξ − y))dσ(y)
Z
≤ Cn 2jn (1 + 2j |ξ − y|)−n dσ(y)


Z
XZ
= Cn 2jn 
(1 + 2j |ξ − y|)−n dσ(y) +
(1 + 2j |ξ − y|)−n dσ(y)
D(ξ,2−j )
D(ξ,2k+1−j )−D(ξ,2k−j )
k≥0
(A)

jn
≤ Cn 2

−j
σ(D(ξ, 2
)) +
X
−nk
2
k+1−j
σ(D(ξ, 2
) − D(ξ, 2
k≥0
(B)

≤ Cn 2jn A2−j(n−1) +

X
A2−nk 2(k+1−j)(n−1) 
k≥0
(C)
= Cn 2jn · 3A2n−1 2−j(n−1) = 3ACn 2n−1 2j .
k−j
))
THE RESTRICTION PROBLEM AND THE TOMAS-STEIN THEOREM
9
Step (A) bounds each integral by the product of the integrand’s supremum and
the measure of its support. Step (B) uses the observation (4.5) to approximate the
measures of the disks. Step (C) evaluates the geometric series on the right and
simplifies the expression.
Now we have the following bound:
(4.6)
j
ˆ
ˆ
\
kKj ∗ f kL2 = kK
j ∗ f kL2 = kK̂j f kL2 ≤ kK̂j kL∞ kf kL2 ≤ Q2 kf kL2 ,
where Q is the constant 3ACn 2n−1 from the computation. We used Plancharel’s
theorem twice to pass to the Fourier transform and back again.
But this means the convolution operator Tj f = Kj ∗ f is bounded L1 → L∞ by
(4.4) and L2 → L2 by (4.6). By Reisz-Thorin,
kKj ∗ f kp ≤ Qθ 2jθ 2−j
where θ =
2
p.
n−1
2 (1−θ)
kf kp0
This is because
1
θ 1−θ
+
=
2
∞
p
and
θ 1−θ
1
1
+
= 1 − = 0.
2
1
p
p
Then
(4.7)
kKj ∗ f kp ≤ Q2/p 22j/p 2−j
n−1
2 (1−2/p)
kf kp0 = Q2/p 2j(
n+1
n−1
p − 2 )
kf kp0 .
The exponent is negative when p > 2n+2
n−1 , as desired.
The remaining step is the simple observation that
X
σ̂ ∗ f = K−∞ ∗ f +
Kj ∗ f.
j≥0
Then combining (4.3) and (4.7), both the left and right sides are bounded for
0
p < 2n+2
n−1 and so kσ̂ ∗ f kp ≤ Ckf kp , which completes the proof.
Remark 4.8. No special properties of the sphere were used in the proof. The relevant
facts were Theorem 3.4 and equation (4.5). The latter only requires the surface to
be a compact submanifold of codimension 1, while the former needs nondegeneracy
at the critical points of the phase functions. This turns out to be equivalent to
having nonzero Gaussian curvature. We will use this in the next section in deriving
Strichartz estimates for the Schroedinger equation.
5. Strichartz Estimates
An application of the Tomas-Stein theorem is the derivation of Strichartz estimates for partial differential equations. These bound the Lp norm of the solution in
terms of Banach space norms of the initial data. They are a powerful tool in dealing with the limiting processes often necessary to answer existence and uniqueness
questions. A simple example is the homogeneous Schroedinger equation:
1
−i∂t u + 2π
∆u = 0
(5.1)
u(x, 0) = f (x)
where u : Rn ×R → R depends on position (x) and time (t), ∂t is the time derivative,
and ∆ is the Laplacian in the position variables only. We can fix time and take the
10
DENNIS KRIVENTSOV
x Fourier transform (which we denote with the usual hat symbol) of the equation,
which gives the ordinary differential equation
−i∂t û(ξ, t) − 2π|ξ|2 û(ξ, t) = 0
(5.2)
û(ξ, 0) = fˆ(ξ)
where we used the property of the Fourier transform that ∂d
xj u(ξ) = (2πiξj )û(ξ) on
the Laplacian. But (5.2) is easy to solve and yields
2
û(ξ, t) = fˆ(ξ)e2πit|ξ| .
Now we just need to take the inverse Fourier transform to get
Z
2
(5.3)
u(x, t) = fˆ(ξ)e2πi(ξ·x+t|ξ| ) dξ.
This can be reexpressed as the inverse Fourier transform of fˆµ, where µ is the
measure in Rn × R given by
Z
Z
φ(ξ, t)dµ =
φ(ξ, |ξ|2 )dξ
Rn ×R
Rn
for φ ∈ C 0 . This is a measure supported on the surface of a paraboloid. By Remark
4.8, the Tomas-Stein theorem applies to ψµ where ψ ∈ C0∞ and is 1 on the unit
= 2 + n4 ,
ball in Rn × R. The paraboloid is n dimensional, so for p ≥ 2n+4
n
(5.4)
k(fˆµ)ˇkLp ≤ CkfˆkL2 (µ) = CkfˆkL2 (Rn )
for f with fˆ supported in the unit ball. The last equality follows from the fact that
integrating a time-independent function with respect to µ is the same as integrating
it with respect to Rn .
The left hand side of (5.4) is kukLp by construction, while the right hand side is
Ckf kL2 (Rn ) by Plancharel, so
(5.5)
kukLp (Rn+1 ) ≤ Ckf kL2 (Rn ) .
Now let fˆ have support in D(0, λ). If we let fλ (x) = f ( λx ) and uλ (x, t) =
(5.1) implies
1
−i∂t uλ + 2π
∆uλ = 0
(5.6)
uλ (x, 0) = fλ (x)
u( λx , λt2 ),
by the chain rule. But fˆλ (ξ) = λ−n fˆ(λξ), which has support in D(0, 1). By change
of variables, kfλ k2L2 (Rn ) = λn kf k2L2 (Rn ) and kuλ kpLp (Rn+1 ) = λn+2 kukpLp (Rn+1 ) . Combining this with (5.5) gives
(5.7)
p
p
p
n
kukLp (Rn+1 ) = λ n+2 kuλ kLp (Rn+1 ) ≤ Cλ n+2 kfλ kL2 (Rn ) = Cλ n+2 λ 2 kf kL2 (Rn ) ,
and the left hand side is Ckf kL2 (Rn ) when p = 2 + n4 . This means that for that
value of p, (5.5) holds for all f ∈ L2 with compact support (uniformly in C). By
the density of compactly supported functions, this means it holds for all f in L2 ,
which is the Strichartz estimate we wanted.
Acknowledgments. It is a pleasure to thank my mentor Andrew Lawrie for his
help in introducing me to the material, refining my topic, and editing this paper.
THE RESTRICTION PROBLEM AND THE TOMAS-STEIN THEOREM
11
References
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[2]
[3]
[4]
Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley. 1999.
Walter Rudin. Real and Complex Analysis. McGraw Hill. 1986.
Wilhelm Schlag. Harmonic Analysis Notes. http://www.math.uchicago.edu/ schlag/book.pdf.
Thomas H. Wolff. Lectures on Harmonic Analysis. American Mathematical Society University
Lecture Series. 2002.